Teichmüller space

Summary

The Teichmüller space $T_g$ parametrises the complex structures on a fixed closed surface of genus $g$. Each point also carries a marking, which remembers the topology up to isotopy. For $g \ge 2$ it is a complex manifold of dimension $3g - 3$ and is homeomorphic to $\R^{6g - 6}$. The mapping class group acts on it properly discontinuously, and the quotient is the moduli space of curves of genus $g$.